Setup and Rule Diagrams

[Dave Killoran]

Setup and Rule Diagram Explanation

This is a Grouping/Linear Combination, Identify the Templates game.

The scenario establishes that nine cookie batches are made each week:

PT42-Dec2003_LGE-G3_srd1.png

These nine batches of cookies are made over five days, Monday through Friday:

PT42-Dec2003_LGE-G3_srd2.png

From the first rule, we can infer that each kind of cookie is baked sequentially:

PT42-Dec2003_LGE-G3_srd3.png

This rule could also be represented in not-block form:

PT42-Dec2003_LGE-G3_srd4.png

This rule creates a large number of Not Laws (O2, O3, P2, P3, and S2, and S3 cannot be made on Monday, O3, P3, and S3 cannot be made on Tuesday, etc) but it is too time-consuming to show all of those Not Laws, and of limited value since they are relatively obvious (2 can’t go before 1, etc).

The second rule states that at least one batch of cookies is made on Monday. Thus, we know (at this point) that at least O1, P1, or S1 must be made on Monday. For some people this rule is a bit of a surprise because they assumed that there was a batch made on each day, but note that nowhere in the scenario is that statement made. Always check the rules to make sure that the variable requirements are clear, and do not assume that certain things must happen just because they usually do in other games.

The third rule indicates that O2 and P1 are made on the same day:

PT42-Dec2003_LGE-G3_srd5.png

This can be a tricky rule. Because the rule involves O2, which is the second batch of oatmeal cookies, this block cannot be made on Monday. Thus, neither O2 nor P1 can be made on Monday, meaning that, at least, O1 or S1 is made on Monday:

PT42-Dec2003_LGE-G3_srd6.png

Because P1 is associated with the block, the block also cannot be placed on Thursday or Friday (if it were, there would be no room for P2, P3, or O3, depending on the day). Thus, the block in this rule must be made on Tuesday or Wednesday, and this is a major inference in the game (after examining the next rule, this dual-option for the block will be used to make templates).

The fourth rule establishes that S2 is made on Thursday, and from the first rule, we can infer that S3 is then made on Friday:

PT42-Dec2003_LGE-G3_srd7.png

Of course, S1 must be made on Monday, Tuesday, or Wednesday.

With each of the rules reviewed in detail, let us look at the two templates created by the block in the third rule:

Template #1: P1 and O2 made on Tuesday

In this template, because O2 is made on Tuesday, O1 must be made on Monday.

PT42-Dec2003_LGE-G3_srd8.png

O3, P2, P3, and S1 are still unassigned, and limited only by the first rule. Thus, this template contains a large number of possible solutions.

Template #2: P1 and O2 made on Wednesday

In this template, because P1 is made on Wednesday, P2 must be made on Thursday, and P3 must be made on Friday. O3 is limited to Thursday or Friday.

PT42-Dec2003_LGE-G3_srd9.png

O1 and S1 are still unassigned, but at least one must be made on Monday. Overall, this template contains a smaller number of possible solutions than Template #1.

[catherinedf]

Hi,

I'm working on the 3rd game of the 2003 December test - bakery making cookies.

I'm not really sure how to set up this game. I put the days of the week as the base and then put the variables as the cookies: 0, P, S. My game set up looks like this:

O1, 02, 03
P1, P2, P3
S1, S2, S3

O2
P1 (box around these two)


___ ___ ___ __S_ __S_
M T W Th Fr
P (with slash)

I missed half the questions on this section. Am I missing some key inferences or just moving too fast?

Thanks!

Catherine

[Nikki Siclunov]

The trick with this game is to realize that you don't need to have a batch of cookie on any of the days, except for Monday and Thursday. Likewise, there is no limit as to how many batches you can have on a given day - you can have all 3 fall on the same day.

The variables are: O O O P P P S S S

Because no two batches of the same kind of cookie are made on the same day, you can sequence the three batches of each kind as follows:

O > O > O
P > P > P
S > S > S

Since the second batch of O is made on the same day as the first batch of P, you can combine the two O and P chains:


O > OP > O

and

O > OP > P > P


As you mentioned, we have S on Thursday and S on Friday. The OP block can only go on one of two days: Tuesday or Wednesday. Based on that, you can make two templates: the OP falls on Tuesday, the first O will be on Monday; if the OP falls on Wednesday, you will have P on Thursday and P on Friday, with the third O on either of these two days. You always have either O or S on Monday, since at least one batch is always made on Monday.

Try the templates approach and see if it works better.

[catherinedf]

Nikki,

Thanks for suggesting templates - it seems a whole lot easier when the OP block can only be on two days.

Thanks for all your help!

Catherine

[deana]

Hi,

I am having a very hard time with these two games. I was able to quickly do the other two problem sets without missing any but got totally stumped with these two.

For the questions 13-18 My base was Mon-Fri and I used OOOPPPSSS to represent the three batches of cookies. What inferences should I be getting and is my set up correct? I can't figure out not laws etc...

Thanks so much I know this is a huge question!

[Jon Denning]

Hi Deana - thanks for the question. For game 3, the first thing I noticed was the unbalanced numbers, and the distribution(s) that creates. It seems at first as though you have a 9 into 5 distribution (9 total batches, 5 days), however it is important to realize that some days could be empty. So in theory it could actually be a 9 into 4 distribution, where one day is cookie-less. Note that 9 into 3 is impossible, as that would require the first, second, and third batch of each type be on the same days as each other, and rule 3 eliminates that possibility.

With that in mind, let's look at the rules/inferences. Monday must have at least one batch, which doesn't really tell you much aside from the fact that you need to keep that in mind as you consider answers (C in 13, for instance). From the third rule we know that PB cannot be Mon, as the first PB is the same day as the 2nd O. So Monday could be S, O, or both. And finally if the 2nd S is on Thursday, then the 3rd S must be Friday.

At this point it just becomes a matter of keeping track of the numbers (so that all 9 batches get made), and using the 3rd rule to dictate the relationship between PB and O.

I hope that helps. If you are still struggling with individual questions feel free to let us know and we'll help you out with those too. Thanks!

Jon

[deana]

Thanks so much Jon! I was able to answer the questions within the 8:45 time frame. I did miss #14 though and I am still a little stumped. I answered C and the answer is A. I can get three with 2 cookies so I don't understand why the answer is 1.

Thanks,
Deana

[Jon Denning]

Well the question is how many days are there where AT MOST two types of cookies could be made. In other words, how many days are there where we can't have all three? We know that we can't have all three on Monday, as the first batch of PB goes on the same day as the second O (so can't have PB Monday, as that would be the first batch and there's not room for O ahead of it). But any of the other four days could have all three batches. So the answer is that only one day has a max limit of two.

Hope that helps!

[cmorris32]

Hi PowerScore!

I was looking at the previous comments and it seems like most people used Monday-Friday as a base. I did not set up the game this way, and I was wondering if having a base of Monday-Friday in this particular instance is better than what I did?

[Adam Tyson]

Whenever you have a variable set that has inherent order to it, like time slots or months or days of the week, that makes the better base than using those elements as moving variables. They can be fixed in place without any room for error or confusion, and the relationships between them are clearly understood. Rules and questions about "before" and "after" work most naturally when the order is used in a fixed left to right (or top to bottom, in some cases) orientation.It becomes easier to see those relationships when it looks like our normal expectations of a calendar or a schedule.

Of course, it is all a matter or preference, and if the way you set it up worked well for you, allowing you to quickly and easily make inferences and answer questions, then who are we to say what's better? You have to do what works best for you!